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set of limit points is closed #1870

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9e257c1
set of limit points is closed
affeldt-aist Mar 3, 2026
c094586
generalization
affeldt-aist Mar 9, 2026
6ac658b
rm
affeldt-aist Mar 9, 2026
b258208
minor improvement
affeldt-aist Mar 9, 2026
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6 changes: 6 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -72,6 +72,12 @@
- in `derive.v`:
+ lemmas `compact_EVT_max`, `compact_EVT_min`, `EVT_max_rV`, `EVT_min_rV`

- in `topology_structure.v`:
+ lemma `limit_pointNE`

- in `separation_axioms.v`:
+ lemmas `limit_point_closed`

### Changed

- in `constructive_ereal.v`: fixed the infamous `%E` scope bug.
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25 changes: 25 additions & 0 deletions theories/topology_theory/separation_axioms.v
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Expand Up @@ -201,6 +201,31 @@ Arguments hausdorff_space : clear implicits.
Arguments accessible_space : clear implicits.
Arguments kolmogorov_space : clear implicits.

Lemma limit_point_closed {T : topologicalType} (A : set T) :
accessible_space T -> closed (limit_point A).
Proof.
move=> accT; rewrite -openC openE/= => a.
rewrite /setC/= limit_pointNE => -[X].
rewrite nbhsE/= => -[U oaU UX] XAa.
rewrite /interior nbhsE/=.
exists U => // x Ux /=.
rewrite limit_pointNE.
have [xa|xneqa] := eqVneq x a.
exists U; rewrite xa; first exact: open_nbhs_nbhs.
by apply: subset_trans XAa; exact: setIS.
exists (U `&` [set~ a]).
apply: open_nbhs_nbhs; split.
apply: openI; first by case: oaU.
by rewrite openC; exact: accessible_closed_set1.
by split => //; exact/eqP.
apply: (@subset_trans _ (A `&` (X `&` [set~ a]))).
by apply: setIS; exact: setSI.
apply: (@subset_trans _ ([set a] `&` [set~ a])).
by rewrite setIA; exact: setSI.
by rewrite setICr.
Qed.
Arguments limit_point_closed {T} A.

Lemma subspace_hausdorff {T : topologicalType} (A : set T) :
hausdorff_space T -> hausdorff_space (subspace A).
Proof.
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9 changes: 9 additions & 0 deletions theories/topology_theory/topology_structure.v
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Expand Up @@ -689,6 +689,15 @@ Proof. by rewrite limit_pointEnbhs; under eq_fun do rewrite meets_openr. Qed.
Lemma subset_limit_point E : limit_point E `<=` closure E.
Proof. by move=> t Et U tU; have [p [? ? ?]] := Et _ tU; exists p. Qed.

Lemma limit_pointNE A a : (~ limit_point A a) =
exists2 X : set T, nbhs a X & A `&` X `<=` [set a].
Proof.
rewrite /limit_point/= -existsNE exists2E; apply: eq_exists => X/=.
rewrite not_implyE -forallNE; congr and; apply: eq_forall => t/=.
rewrite and3E not_andE (propext (rwP negP)) negbK implyE orC.
by rewrite -(propext (rwP eqP)).
Qed.

Definition isolated (A : set T) (x : T) :=
x \in A /\ exists2 V, nbhs x V & V `&` A = [set x].

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